The title says most of it really. How constrained are you in an academic career based on the topic of your PhD thesis? Is it plausible for someome to be researching in an area of math vastly different from their thesis topic later in an academic career, or does it largely determine the scope of research you will be funded for and publish in the future?
I am interested in this as a general question, but I will add some details about my situation that has inspired the question. I am a graduate student at the moment, working in the intersection of homological algebra, triangulated categories, and algebraic geometry. Primarily within sections 18E, 18F, 18G of the AMS subject classification. I'm interested in the future in getting into more pure algebraic geometry research. Is this plausible? It may seem like a stupid question, but my understanding is that there is a much steeper technical learning curve. My current exposure to algebraic geoemtey is roughly a graduate course, so the bulk of chapters 2 and 3 of Hartshorne.